Abstract. We show that the members of a large class of unbalanced four-manifold trisections are standard, and we present a family of trisections that is likely to include non-standard trisections of the four-sphere. As an application, we prove a stable version of the Generalized Property R Conjecture for c-component links with tunnel number at most c.

We prove a theorem which bounds Heegaard genus from below under special kinds of toroidal amalgamations of $3$-manifolds. As a consequence, we conclude $t(K_1\# K_2)\geq \max\{t(K_1),t(K_2)\}$ for any pair of knots $K_1,K_2\subset S^3$, where $t(K)$ denotes the tunnel number of $K$.Comment: 22 pages, 6 figure

If the tunnel number of a knot K is denoted t(K), a pair of knots K 1 , K 2 is said to be subadditive if t(K 1 ) + t(K 2 ) > t(K 1 #K 2 ). In [11] Scharlemann and Schultens defined the degeneration ratio to be d(K 1 , K 2 ) = 1 − t(K1#K2) t(K1)+t(K2) , and proved that d(K 1 , K 2 ) ≤ 3/5. However, the highest known degeneration ratio known for a pair of knots is just 2/5. We use free decompositions to construct links which experience degeneration approaching 3/7 when the connect sum is taken with certain knots. These links can be modified to yield a family of knots whose members we conjecture to have the same property.

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We classify a large class of "unbalanced" 4-manifold GK-trisections, which are a slight generalization of 4-manifold trisections defined by Gay and Kirby in [4].